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I am studying DFA's, but I've been struggling to find an explanation of what does the final state mean? I know that it is indicated by double circle on the graph, but what does it imply?

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    $\begingroup$ At some point, we learn to live without asking too many "what is _?" questions. Math is not really about natural things, things that "are". $\endgroup$ – André Souza Lemos Nov 26 '16 at 13:16
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If you want to know what a mathematical object is, you need to read its definition. So, I imagine you've seen the definition of finite automata, and it has something called a "set of final states", and the definition probably doesn't say what that means.

OK, that means that, for now, the author is just giving a name to this set of states. They surely wouldn't bother naming something if it was never going to be used, so we need to keep reading. The very next thing you read is probably a desciption of how an automaton processes its input by moving from state to state as it reads the characters. That doesn't seem to involve the concept of final states, so we keep reading some more. The next thing is probably a definition of what inputs are accepted by the automaton. That definition does involve final states, and it tells you exactly what their role is.

The moral of the tale is that, if you want to know what a mathematical object is, you need to read its definition. If you want to know what it's for, you might have to read a bit more, after its definition. (A good writer will warn you if the point of something isn't going to be revealed for a while.) If you want to know why anyone should care about the object, you might have to do a lot of reading, or find someone (or somewhere! ;-) ) to ask questions.

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When the automaton has processed the entire input, it accepts the input if (and only if) it s in a state marked as final.

It is really no more complicated than that.

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